Class: Pre-Algebra, Algebra, Statistics, Discrete Math

BUYING TIME

Cindy Boyd Anthony Ice Donna Osborn Margaret Sanders

Materials: Poll sheets, tally sheets, calculator, background information, practice problem, student project sheet.

Goals: After gathering information on preferences, examine the different methods for determining a winner - Plurality, Borda Count, Run-Off, Sequential Run-Off, and Condorcet. To enable students to become familiar with oral and written presentations in Discrete Math.

Time Required: Approximately 2-4 days. One day to teach methods, if needed. One day to tally the data, then time either in or out of class to apply the decision making methods, make the decisions, and write the report. One day for oral presentations.

Background: Knowledge of methods of voting, frequency tables and survey techniques. Following this page are a series of pages illustrating these different methods with different soft drinks.

Setting: A client has come into your advertising agency with a new theme park for fun-loving young adults. They want to advertise during one prime time situation comedy.

Problem: Take a preferential survey of the top three situation comedies. Use Plurality, Borda Count, Run-Off, Sequential Run-Off, and Condorcet methods to determine winners. Then make a final choice. Each advertising agency should submit a data sheet and a written recommendation which defends and explains their method for determining when to advertise the new product. Also, each agency will orally present their report to the client.

Evaluation: One grade will be given to the entire group. Point values will be assigned for the written report and presentation. Points for the written report should be divided among the description of the problem, the list of assumptions, the data sheet with the results of the survey, the results of the decision-making processes, the recommendation for the client, and the explanation of the decision.

Extension:

1. For a more challenging project, increase the survey to include four TV shows.

2. Call local TV stations to find the cost for advertising during the shows surveyed.

3. Choose a specific night or time slot.

4. Change the product and/or target audience.

Teacher Notes:

1. To choose the shows for the survey: Have each student put his/her favorite prime time situation comedy on a slip of paper. Place slips in a container or a box. Draw out three slips (with different shows); these will be the three shows used in the survey. If two classes are doing the project simultaneously, use the same shows for both classes.

2. Survey at least 2 classes (50 students).

3. Depending on the level of the class, the background concept sheets could be run for all of the students. The background concept sheets are explanations of the different decision making methods.

Funded in part by the National Science Foundation and Indiana University 1995

4. The project can be scaled up or down to match the level of the class by increasing/decreasing the number of methods used to choose a winner.

5. The oral presentation should present the written report to the class. It should not be read.

6. Enclosed is a student tally sheet with six places for tallies; these are for the six possible rank orders. If four choices are given, then there are 24 possible rank orders.

Funded in part by the National Science Foundation and Indiana University 1995

PREFERENTIAL VOTING

Consider this set of preferences for soft drinks. Each column represents the rank order, while the number below indicates the number of people voting that preferential order.

Pepsi / Diet Coke / Dr. Pepper / Coke / Sprite / Sprite
Coke / Sprite / Diet Coke / Dr. Pepper / Diet Coke / Dr. Pepper
Sprite / Coke / Sprite / Sprite / Coke / Coke
Dr. Pepper / Dr. Pepper / Coke / Diet Coke / Dr. Pepper / Diet Coke
Diet Coke / Pepsi / Pepsi / Pepsi / Pepsi / Pepsi
18 / 12 / 10 / 9 / 4 / 2

Some methods of determining the winner:

1. Plurality

2. Borda Count

3. Run-Off

4. Sequential Run-Off

5. Condorcet

When a tie occurs for any of the methods, use the other methods to make a decision.

The Plurality Method

The winner is the choice that receives the most first place votes.

Diet Coke = 12 Pepsi = 18 Coke = 9 Dr. Pepper = 10 Sprite = 4 + 2 = 6

The plurality winner is Pepsi. The plurality winner is not always the majority winner. Note that the plurality winner is ranked last by a majority of the voters.

Funded in part by the National Science Foundation and Indiana University 1995

The Borda Method

If there are N choices on the ballot, the Borda method awards N points for a first-place ranking, N-1 points for a second-place ranking, etc.

In order, the totals for Diet Coke, Pepsi, Coke, Dr. Pepper, and Sprite:

Diet Coke = 18(1) + 12(5) + 10(4) + 9(2) + 4(4) +2(2) = 156

Pepsi = 18(5) + 12(1) + 10(1) + 9(1) + 4(1) +2(1) = 127

Coke = 18(4) + 12(3) + 10(2) + 9(5) + 4(3) +2(3) = 191

Dr. Pepper = 18(2) + 12(2) + 10(5) + 9(4) + 4(2) +2(4) = 162

Sprite = 18(3) + 12(4) + 10(3) + 9(3) + 4(5) +2(5) = 189

Coke wins.

Funded in part by the National Science Foundation and Indiana University 1995

The Run-Off Method

The Run-Off method conducts a new election between the two choices with the most first-place rankings.

Diet Coke = 12; Pepsi = 18; Coke = 9; Dr. Pepper = 10; Sprite = 4 + 2 = 6

The Run-Off is between Diet Coke and Pepsi. Eliminate all but Pepsi and Diet Coke. Next consider each of the preference schedules which were eliminated. Dr. Pepper (10) should be awarded to Diet Coke because Diet Coke was rated above Pepsi in that preferential list. Coke (9) will be awarded to Diet Coke. Sprite (4) will be awarded to Diet Coke. Sprite (2) will be awarded to Diet Coke.

Diet Coke = 12 + 10 + 9 + 4 + 2 = 37

Pepsi = 18

Diet Coke wins.

Funded in part by the National Science Foundation and Indiana University 1995

The Sequential Run-Off Method

The sequential Run-Off method eliminates only one choice and then conducts a new election. This process is repeated until one choice remains.

Diet Coke = 12; Pepsi = 18; Coke = 9; Dr. Pepper = 10; Sprite = 4 + 2 = 6

Sprite is eliminated first. Sprite=s four votes in column five are awarded to Diet Coke because Diet Coke was the second choice of those four people. Sprite=s two votes in column six are awarded to Dr. Pepper because Dr. Pepper was the second choice of those two people. Now the vote totals are changed as shown below.

Diet Coke = 12 + 4 = 16

Pepsi = 18

Coke = 9

Dr. Pepper = 10 + 2 = 12

Now the new low, Coke, is eliminated and Coke=s nine votes are awarded to Dr. Pepper because Dr. Pepper was the second choice of those nine people. Now the vote totals are changed as shown below.

Diet Coke = 16

Pepsi = 18

Dr. Pepper = 12 + 9 = 21

Now the new low, Diet Coke, is eliminated and Diet Coke=s 16 votes are given to Dr. Pepper because the second choice, Sprite, and the third choice, Coke, have already been eliminated. Now the vote totals are changed as shown below.

Pepsi = 18

Dr. Pepper = 21 + 12 = 33

Now the new low is eliminated and Dr. Pepper wins. But it is interesting to note that 43 people ranked Coke above Dr. Pepper originally and 12 people ranked Dr. Pepper above Coke.

Funded in part by the National Science Foundation and Indiana University 1995

The Condorcet Method

The Condorcet method awards the election to the choice that can beat each of the other choices in one-on-one contests.

To examine the data for a Condorcet winner, compare each choice with every other choice. First put an X in each box which compares a choice to itself. Then compare Diet Coke with Pepsi. Diet Coke is ranked higher than Pepsi on columns 2, 3, 4, 5, and 6; therefore Diet Coke gets the votes from those columns (12+10+9+4+2=37 votes). Pepsi is ranked higher than Diet Coke on column 1; therefore, Pepsi gets the votes from that column (18 votes). So Diet Coke wins and we place a W in the Diet Coke row, Pepsi column. Now we compare Diet Coke to Coke. Diet Coke is ranked higher than Coke in columns 2, 3, and 5; therefore Diet Coke get the votes from those columns (12 + 10 + 4 = 26 votes). Coke is ranked higher than Diet Coke in columns 1, 4, and 6; therefore, Coke gets the votes from those columns (18+ 9 + 2 = 29 votes). So Diet Coke loses and we place an L in the Diet Coke row, Coke column. Since Diet Coke does not beat all of the other choices, Diet Coke can not win. Now we would begin the same process with Pepsi.

Diet Coke / Pepsi / Coke / Dr. Pepper / Sprite
Diet Coke / X / W / L / L / L
Pepsi / L / X / L / L / L
Coke / W / W / X / W / L
Dr. Pepper / W / W / L / X / L
Sprite / W / W / W / W / X

The winner is Sprite because it obtains a majority over each of the other choices. If no one wins against all of the other choices, then there is no winner. This is called a paradox.

Funded in part by the National Science Foundation and Indiana University 1995

Practice Problem

Student Worksheet

Here are some survey results for you to analyze using the different methods. The school cafeteria is considering three different possibilities - pizza, hamburger, or hot dog - for lunch. Only one of the three will be served. Here are the results of the survey. Use all of the different methods to determine the lunch choice.

Pizza / Pizza / Hot Dog / Hot Dog / Hamburger / Hamburger
Hamburger / Hot Dog / Pizza / Hamburger / Pizza / Hot Dog
Hot Dog / Hamburger / Hamburger / Pizza / Hot Dog / Pizza
13 / 18 / 12 / 17 / 13 / 17

Funded in part by the National Science Foundation and Indiana University 1995

ANSWERS TO PRACTICE PROBLEM

PLURALITY: Pizza

BORDA COUNT: Hot Dog

RUN-OFF: Hamburger

SEQUENTIAL RUN-OFF: Hamburger

CONDORCET: Hot Dog

Funded in part by the National Science Foundation and Indiana University 1995

VECTOR AD AGENCY

You are a member of a creative team at the Vector Ad Agency, Athe agency with magnitude and direction@. SumWare Company is offering your firm a chance to market their newest theme park, CalcLand. SumWare wants to purchase ad-time during the most popular prime time situation comedy. It is important to Vector Ad Agency to snag the SumWare account from a rival company, Shakespeare Ltd. The SumWare account would generate over $2.1 million for your agency. Your boss has assigned the account to your creative group and to another group in the firm. Your boss will use the written and oral presentations to decide which group gets the $25,000 bonus.

Your group=s plan of action is:

a) To conduct a preferential survey of the top three situation comedies

______, ______, and ______.

b) To use Plurality, Borda Count, Run-Off, Sequential Run-Off, and Condorcet to determine

winners.

c) To make a final choice using the results of your decision-making process.

Your creative team will

* submit a written report which contains

* a description of the problem

* a list of assumptions

* a data sheet with the results of the survey

* results of the decision-making processes

* a recommendation for the client

* an explanation of your decision

* give an oral presentation which covers everything in the written report

Funded in part by the National Science Foundation and Indiana University 1995

Rank the 3 shows in order with 1 as your favorite. / Rank the 3 shows in order with 1 as your favorite.
______/ __ / ______/ __
______/ __ / ______/ __
______/ __ / ______/ __
Rank the 3 shows in order with 1 as your favorite. / Rank the 3 shows in order with 1 as your favorite.
______/ __ / ______/ __
______/ __ / ______/ __
______/ __ / ______/ __
Rank the 3 shows in order with 1 as your favorite. / Rank the 3 shows in order with 1 as your favorite.
______/ __ / ______/ __
______/ __ / ______/ __
______/ __ / ______/ __
Rank the 3 shows in order with 1 as your favorite. / Rank the 3 shows in order with 1 as your favorite.
______/ __ / ______/ __
______/ __ / ______/ __
______/ __ / ______/ __

Funded in part by the National Science Foundation and Indiana University 1995